To illustrate this, I have superimposed two exposures in an animated GIF. They were taken with the Lumix G 14mm f/2.5 at f/4, 1/60s, ISO 200. Normally, one would think that 1/60 second is safe for handholding a shot with a wide angle lens, however, since each full exposure takes 1/10 second, any movement during the exposure will result in a skewed image:
Looking at only one of the two exposures above, one might not notice any problems. However, when seeing both, it is clear that at least one of them, probably both, are not geometrically correct. So, is this a problem? When holding the camera reasonably still during the exposure, and not photographing very square objects, it is no issue. If you critically need rectilinear images, then you are better off using the normal mechanical shutter.
In artificial light
For about a century or so, people have been using incandescent light bulbs for electronic indoor lightning. Even when used on alternating current (AC), the light is stable. Since the filament is heated, it emits light also when the alternating current is at zero.
However, traditional incandescent light bulbs are now being replaced with the energy saving fluorescent light bulbs. They tend to flicker at 100Hz (in Europe) or at 120Hz (in the US). The lights don't flicker at 50Hz and 60Hz, as you might expect. This is since during each period, the electrical current reaches two peaks, see the illustration below:
As the Panasonic GH3 takes 1/10s to record an image in electronic shutter mode, the flicker can be a problem. Any shutter speed faster than 1/100 second will be a problem.
The fastest shutter speed which is safe, is 1/100s (or 1/120s if you are in a 60Hz country, like the US). But don't think that a slower shutter speed will be ok. A slightly slower shutter means that you capture two of the peaks at some points, giving banding again. Below are the exposures at 1/100s and 1/80s. Only 1/100s is ok, since I am on a 50Hz power grid:
On a 50Hz power grid, the safe shutter speeds are 1/100s, 1/50s, 1/25s, and so on. In the US, you should aim for 1/120s, 1/60s, 1/30s, and so on. When photographing indoors, you are likely to use a slow shutter speed, so this is not likely to be a big problem.
We can also see that there are ten bands of grey periods in my pictures, indicating that the rolling shutter exposure takes around 1/10s. This is exactly what I concluded using my Technic LEGO contraption earlier!
What is a negative side effect to some, can be a creative possibility for others. The rolling shutter effect can be used creatively, to create interesting images. One way is to photograph moving objects, to make it lean. An obvious choice can be a moving car, 1/1600s, ISO 400:
If you want the car to lean the other way, then you must hold the camera upside down!
Also, when photographing vibrating items, you can get fun effects. Here, I have photographed a bass player's hand:
The picture was taken at with the Lumix X 45-175mm lens at 175mm f/5.6, 1/800s, ISO 200. In retrospect, I should have set the ISO higher, to achieve a faster shutter speed, and better defined green string, with less motion blur.
When taking a picture like this, it is important not to move the camera during the exposure, otherwise, you get the negative rolling shutter effects. Here, my hand has apparently shaken a bit during the exposure, giving a slightly S-shaped guitar:
It is also important to have the vibrating string vertical. If it was horizontal, you would get a different effect.
Here is a guitar player as well:
On the positive side, using the electronic shutter reduces the vibrations. This can be good when using a long lens on a tripod. The mechanical shutter can start a vibration, which will reduce the sharpness of the image. Now, this is rather theoretical, so let's try to see an example. I put the camera on a tripod, used a Lumix G 100-300mm f/4-5.6 at 300mm f/5.6, 1/30s, ISO 200, using a 10 second shutter delay to avoid vibrations. Here are 100% crops from the resulting images when using the electronic shutter (left) and the mechanical shutter (right):
As you can see, the image taken using the electronic shutter is slightly sharper than the one taken using the mechanical shutter. This is probably due to the mechanical shutter setting off a small vibration, reducing the sharpness when using a long focal length. However, the difference is rather subtle, and would probably not make much of a difference in a real life example. If you have an even longer lens, then this might be more important to look into. There are, at this time, no native lenses longer than 300mm. But it is possible to use legacy lenses on adapters.
Comparison with other systems
The Nikon 1 mirrorless cameras were designed with electronic shutter in mind from the start. The Nikon 1 S1 10MP camera has a 1/60s readout, six times faster than that of the Panasonic GH3. The Nikon 1 J3 14MP camera is even better, with an electronic shutter capable of reading the whole image during 1/80s.
Only the Nikon 1 V2 camera features a mechanical shutter at all, usable when you want to be sure the image comes out without any rolling shutter artefacts. Further, the camera can take full 14MP images at a staggering 60FPS rate, including RAW, using the electronic shutter, opening up for very interesting uses.
From the Sony NEX line, some of the recent cameras, e.g., NEX-5R and NEX-6 have an electronic front curtain shutter. This means that the mechanical shutter is not closed prior to the start of the exposure. The only mechanical shutter is the one which closes when the exposure is stopped, the rear curtain. The advantage is obvious: It reduces the vibrations before the exposure commences, and also reduces the number of mechanical movements overall, while still avoiding the rolling shutter artefacts discussed above.